![]() The result is surprising because the curve actually dips below the right end point. The path along which a ball would roll in minimum time under the influence of gravity. The finite-length curve (x(s), y(s)) is parametrized over, so Use the endpoints to solve for the unknown constants K and _C1 The parameter s is not equivalent to time t, so we must solve for the value of s at which ![]() XInit := 0 yInit := 1 圎nd := 1 yEnd := 1/2 Our first example of a variational problem is the planar geodesic: given two points lying in a. Specify the endpoints of the curve: (0,1) and (1, 1/2) ODE1 := remove(has, EL, diff(y(x),x,x)) Ĭompute y(x) as a parametric curve (x(s), y(s)) using the dsolve command with the parametric option. This returns a set of ODEs.ĮL := EulerLagrange( fallTime, x, y(x) ) We then use the EulerLagrange function to compute the Euler-Lagrange equations for this functional in terms of y(x) and its derivatives. This is found in standard textbooks on classical mechanics.įallTime := sqrt( (1+diff(y(x),x)^2)/(2*(yInit-y(x))) ) To begin with, we show that the critical points of the length functional in the space of piecewisesmooth curves joiningpandqare exactly the geodesic segments, up to reparametrization. The Brachistochrone problem can be stated as follows: Given two endpoints in the plane, find the curve y(x) between them such that a ball of unit mass rolls along the curve under the influence of gravity in minimum time.įirst we write down the falling time over an infintesimal distance dx in terms of y(x) and yInit, assuming the gravitational constant is 1. In our case, the apparatus of classical calculus of variations can be applied tocarry out this program. The VariationalCalculus package automates the construction and analysis of the Euler-Lagrange equation. The Euler-Lagrange equation is easy to write down in general but notoriously difficult to write down and solve for most practical problems. Such problems can often be solved with the Euler-Lagrange equation, which generalizes the Lagrange Multiplier Theorem for minimizing functions of real variables subject to constraints. Find the shape of a soap film having minimum surface area spanning a given wire frame. ![]()
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